Yes, but do you have the wherewithal to produce that equipment and to fuel the process? How do you compress wood gas? More importantly, perhaps, why would you want to even if you could? To make an experimental IC vehicle perhaps, but not in order to revolutionise society's personal transport. That would take a whole lot of wood for an indefinite time, while perfectly nice horses could haul carriages all day for the price of some grazing and a drink of water.
Wherewithal? Of course they'd have the wherewithal. For the same reasons that James Watt built an steam engine in a world that had been getting along fine without them. To do some form of work that makes things easier. In a pre-industrial-revolution world, there were
absurdly large horse teams driving stationary engines that did jobs like: pumping water out of agricultural fields; likewise for mines; grinding grain; threshing; sawing wood. James Watt's first engines, which were incredibly successful, were
stationary engines for industrial work (water pumps), not cars for "revolutionizing personal transport". Yes, a world without fossil fuels will still have the wherewithal to invent labor-saving devices for pumping water, and then for improving those devices to make them more efficient.
The internal combustion engine was, originally, an incremental efficiency-upgrade over the steam engine. Nikolas Otto's first ICs were (a) stationary, used to winches farm machinery across fields, and presumably replaced coal-powered steam engines for that use; and (b) powered by coal gas, which is very nearly a one-to-one replacement for well-prepared wood gas. Why would Otto have the wherewithal to put a non-compressible gas into a high-tech engine, to do a job that horses could do just fine? Well, the fact is that he did.
Let's also look at the issue of wood power. Wood has something like
75% the energy density of coal, which is great---we're not talking about a world powered by potato batteries, we're talking about "the fuel hopper needs to be a little bigger and the fireman to shovel a little harder". Steamships, railroads would work just fine. Wood was indeed harvested, on coal-like industrial scales, as a power source. For example, the 19th century glass industry of Upstate New York (Corning, etc.) were entirely wood-fired (there was no coal in the area), and harvested this renewable fuel on an industrial scale (although they simply mined it out and didn't renew). In late medieval England, woodlands were coppiced and harvested sustainably for charcoal production for centuries.
"Revolutionize personal transport"? Like, cars? First: the world got a long ways into the industrial revolution, and a long ways into a recognizably-modern-looking world, as soon as we had railroads and steamships. Second: Germany got through WWII using mostly Fischer-Tropsch gasoline (from coal), and they managed to run plenty of tanks, cars, trains, etc. on this fuel. Yes, expensive. Yes, a lot of industrial effort per gallon, compared to fossil petroleum. But not so expensive that nobody would bother. That was coal-based, but there is not THAT big a difference between coal-->syngas--->FT liquids and coal-->wood gas-->FT liquids. Factor of 1.5, factor of 2?
a) Without fossil fuels, the world would not have expanded to 6 billion people. It'd have fewer, being resource-constrained. (Is that "fewer because Malthusian starvation killed everyone"? Or is that "fewer because <some other control valve>"? I don't know.) I don't know the world's carrying capacity for coppiced wood. It's probably quite large---not "six billion people with their own cars" large, but it's still large.
b) Without fossil fuels, the world would probably not have 1 private car per 6 people. It'd have fewer, because all of the fuel options are more expensive in this world. (But world gas prices already vary by a factor of 10---from $1/gal in Venezuela to $10/gal in Norway---without making either extreme unrecognizable.) Fine.
c) Without fossil fuels, the world wouldn't have cheap airplane travel. It'd have expensive airplane travel with expensive biofuels. That's fine---for most of the history of air travel, air travel
was expensive (they were expensive high-maintenance planes, carrying few passengers, etc.). It's easy to say today, "Good lord, if aviation fuel tripled in cost, it'd hardly be worth having the industry at all." But early aviation was indeed something north of 3x as expensive as modern aviation.
Here's a (now) $600 round trip from Detroit to Cleveland in 1923.
People do pretty well at exploiting the resources they have. If the resources they had were "trees", I suspect that they'd get to a world where the global tree resource was pushed right to the limit of what it can do. Exactly how it gets there, and how that gets distributed, giving what lifestyle to a population how big? I don't know. But wood can do much more than horses can; and most of what coal can; and a reasonably fraction of what petroleum can. So there you go.
